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International Journal of Management, IT & Engineering Vol. 8 Issue 10, October 2018,
ISSN: 2249-0558 Impact Factor: 7.119
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285 International journal of Management, IT and Engineering
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MCB-DEA: A modified approach for
benchmarking
Reshampal Kaur1
Dr. H.S. Bhatti2
Dr. Monika Aggarwal3
Abstract:Review of existing literature on the study suggests that though non-parametric
technique Data envelopment analysis (DEA) can be used to arrive at benchmarks but it has its
own limitations, like non-inherent benchmarks, more than one benchmark and different
benchmarks for each year, which makes it difficult for an inefficient bank to decide that which
benchmark should be followed for feasible improvement. Thus, there is a need to identify a new
method which can help in ascertaining benchmarks based on multi period analysis for improving
efficiency. The present study endeavours to propose a model which has been named as MCB-
DEA model. The same has been illustrated using a data base of sixteen years for twenty five
public sector banks operating in India. It was found that the suggested model helps in identifying
nearest benchmark, falling in same cluster so as to gradually improve efficiency.
Keywords:Efficiency, Data envelopment analysis, Malmquist TFP Index, Benchmarking,
Cluster analysis.
1 Research Scholar, Panjab Technical University, Jalandhar, India,
2 HOD & Professor, BBSB Engineering College, Fatehgarh Sahib, India
3 Associate Professor, UIAMS, Panjab University, Chandigarh, India, Postal
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I. Introduction and Review of Literature
For every DMU to be a quality DMU and to achieve next level of efficiency, benchmarking is
required. To benchmark is to compare performance against a standard [1] with the objective to
improve performance [2]. It is based on the premise that “why reinvent the wheel when I can
learn from someone who has already done it?” [3].Benchmarking which is also known as process
benchmarking, internal benchmarking and industry benchmarking [4]is a technique that initially
got popular in Japan but considering its huge potential [5] in large firms to small businesses and
public as well as semi-public sector [5,6,7,8,9] it gained popularity worldwide [10].
Particularly for benchmarking in banking sector, DEA has been widely
used[11,12,13,14,15,16,17,18,19,20]but it has some limitations too. The benchmarks provided
by DEA for inefficient DMUs are sometimes largely different in performance behaviour[21],
which provides the target performance way too difficult to achieve. Secondly, presence of more
than one benchmark creates confusion regarding choice of optimum DMU to be followed.
Literature also advocates the use of Malmquist TFP index on DEA results to analyse changes in
the efficiency levels in a multi-period environment [22,23,24,25,26,27,28,29,30] but it does not
provide any information on benchmarks.Some researchers have also used cluster analysis in
combination with DEA to explore relationships between data points in a single time period
[10,31,32,33,34].
As evident, studies have made an attempt to calculate benchmarks based on DEA but none of
these have used clusters on Malmquist TFP analysis to arrive at benchmarks considering multi
period data. Hence the present study proposes to add value to existing literature by proposing a
new Malmquist clusteringbenchmarking model based on DEA (MCB DEA) to calculate
benchmarks using cluster analysis with DEA based Malmquist TFP index.This paper proposes
MCB-DEA model and tests the same on 25 public sector banks operating in India considering
temporal data from the year 1998 to the year 2013.
II. Model Formulation
The proposed MCB DEA model has three steps. In the first step, Charnes, Cooper and Rhodes
(CCR) model of DEA is used for efficiency evaluation of each DMU, for each time period,
separately.
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Following [35],Consider DMUj , (j=1,2,….., n ) using input vector Xj = (x1j , x2j ,… . . , xmj ) to
produce output vector
Yj = (y1j , y2j ,… . . , ysj ) for X j ≥ 0, Yj ≥ 0
For input weights vector V = (v1, v2 ,… , vm ) and output weights vectorU = (u1 , u2 ,… , us)
each DMUk has an optimization problem
Maximize θ = u1y1k + u2y2k + ⋯+ usysk
s. t. v1x1k + v2x2k + ⋯+ vm xmk = 1
u1y1j + u2y2j + ⋯+ usysj ≤ v1x1j + v2x2j + ⋯+ vm xmj for all j = 1,2,… , n.
v1, v2 ,… , vm ≥ 0; u1 , u2 ,… , us ≥ 0….(*)
Corresponding to k = 1,2,… , n (*) gives a set of „n‟ optimization problems. Each problem is
then solved for obtaining values of most favourable input weights v1, v2 ,… , vm and output
weights u1 , u2 ,… , us for each corresponding DMU.
In Step (2)of the proposed MCB DEA model,change in efficiency behaviour of each DMU, is
analysed over the entire time period, in a multi-period environment, by applying Malmquist TFP
index, on the efficiency scores given by DEA for each time period, in Step (1).
The Malmquist TFP index, which measures the change in productivity of a DMU, between two
data periods t1 and t2 , by calculating the ratio of the distances of each data point relative to a
common technology.Following [36], for a firm, at time ,St1 be the production set,then an
output distance function is defined as
Dt1 xt1 , yt1 = inf θϵ R xt1 ,yt1
θ ϵ St1
}…… . (1)
where,Dt1 xt1 , yt1 ≤ 1; with Dt1 xt1 , yt1 = 1 iff DMU is efficient and
further increase in output yt with same input xt isnot possible
Also, Dt2 xt2 , yt2 = inf θϵ R xt2 ,yt2
θ ϵ St2
}…… . (3)
To compute Malmquist productivity index, we define
Dt1 xt2 , yt2 = inf θϵ R xt2 ,yt2
θ ϵ St1
}…… . (4)
Where Dt1 xt2 , yt2 gives the maximum proportional change in outputs yt2 with same inputs xt2 ,
at time t1.
and Dt2 xt1 , yt1 = inf θϵ R xt1 ,yt1
θ ϵ St2
}…… . (5)
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Where Dt2 xt1 , yt1 gives the maximum proportional change in outputsyt2 with same inputs xt2 ,
at time t2.
[37] had defined Malmquist productivity index with reference to the technology of initial period,
t1 as
Mt1 =D t1 xt2 ,yt2
Dt1 xt1 ,yt1 ……...… (6)
Or alternatively, with reference to the technology of final period, t2 as
Mt2 =D t2 xt2 ,yt2
Dt2 xt1 ,yt1 ………… (7)
To avoid an arbitrary choice of reference technology, [36] defined the Malmquist productivity
index of TFP, between periods t1 and t2 ; t1 < t2 , as the geometric mean of Mt1 and Mt2 ,
M xt2 , yt2 , xt1 , yt1 = Dt1 xt2 ,yt2
Dt1 xt1 ,yt1
Dt2 xt2 ,yt2
Dt2 xt1 ,yt1
1
2……. (8)
Equation (8) can also be written as
M xt2 , yt2 , xt1 , yt1 =Dt2 xt2 , yt2
Dt1 xt1 , yt1
Dt1 xt2 , yt2
Dt2 xt2 , yt2
Dt1 xt1 , yt1
Dt2 xt1 , yt1
1
2
… . . 9
Equation (9) is the decomposition of the Malmquist productivity index into two factors. The first
factor outside the bracket represents the efficiency change (or catching-up effect) component and
the second factor, with the bracket, represents the technological change (or Innovation). Thus,
for constant returns to scale,TFP change = Change in Efficiency × Change in Technology …
(10)
In step (3) of the proposed MCB DEA model,Cluster Analysis is used to divide the DMUs into
different clusters, on the basis of their overall Malmquist TFP index and its components found in
Step (2).Here Hierarchical clustering is used to find appropriate number of clusters and K-
meansclustering is used to find homogeneous clusters. K-means clustering, algorithm is a
simple clustering technique [38]. It is considered as a top-rated data mining algorithm for its
simplicity and vast application areas [30,39].In order to perform k-means clustering, the number
of clusters „k‟ is chosen, means of these clusters (centroid) are computed, the distance of each
object to the centroids is determined using some distance measure. Then the objects are grouped
based on minimum distance and new cluster seeds are computed. The process is repeated until
the centroids no longer change or convergence is reached.The objective function of K-means
algorithm is defined as a minimization function:
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J = D(xi
C
j=1
, cj
N
i=1
)
Where N is the number of objects, C is the number of clusters and D is the measure of distance
between pointsxi and cluster mean cj .
III. Data Base
For the testing of the proposed MCB-DEA model, the data related to 25 public sector banks
operating in India,for the time period, starting from the year 1998 till 2013, were collected from
the Statistical Tables Relating to Banks in India, published by the Reserve Bank of India. The
inputs considered for analysis are Owned funds, Deposits, Borrowings and Wage bills. Whereas,
outputs have been taken as Spread and Other income. Table 1 gives the list of public sector
banks, chosen for study, considered as DMUs. Although there are total 27 public sector banks,
operating in India, but for the purpose of uniformity in data, the IDBI Bank and the Bhartiya
Mahila Bank were eliminated as these banks were formulated in the year 2011 & 2015
respectively. Few banks had merged in the year 2015 onwards, So the data till 2013 have been
considered.Table 2 gives the detailed description of the selected variables and Table 3 gives the
descriptive statistics of the data related to these variables.
As per [40], for accuracy in DEA results, the number of DMUs should be greater than three
times of total number of input variables and output variables. In present study, there are 400
observations (25 DMUs * 16 years) and the total number of input-output variables is six (4+2).
Thus, this study observes well the property of minimal number of DMUs.
IV. Empirical Findings
This section presents the results of various steps of the proposed MCB-DEA model. In first step,
a non-parametric, input oriented CCR model of DEA, assuming constant returns to scale is
applied, separately for each financial year, on the data collected for the public sector banks under
study, with the objective to find comparative efficiency level of each public sector banks in each
year. If DEA efficiency score percentage is equal to one hundred then, public sector bank is
identified to be efficient for that year andinefficient, if its DEA efficiency percentage is less than
hundred.
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The results of CCR model of DEA are listed in Table 4, which gives the efficiency level of each
bank under study for each year column wise. The result shows that during all these sixteen years
under study, out of total 400 observations, 204 are for efficient banks and 196 are for inefficient
banks.
In the second step, change in efficiency behaviour of each public sector bank, for each year
based on previous year, is analysed by applying Malmquist TFP index, on the year wise
efficiency scores given by DEA, in table 4, to further find overall TFP change during the whole
sixteen years time period.Table 5 gives the annual average changes of Total Factor
Productivity(TFPCH) and its decomposition into efficiency change (ECH) and technical
efficiency change (TCH), for each DMU. It is observed that over the entire period of study, out
of total 25 PSBs, 20 have shown an increased TFPCH on average annually. Among these 20
banks, 5 banks have shown improvement in ECH and TCH both, although rate of improvement
in ECHis far less than improvement in TCH, with an exception of The Central Bank of India,
which has same rate of improvement in both factors i.e. 1.3 percent.Three banks have shown no
improvement in ECH but have considerable rate of improvement. Twelve banks have faced a
decline in ECH, but even then, high rate of growth of TCHof these banks has resulted into,
growth of TFPCH.
In third step, to assess the cluster tendency of the dataset, Hopkins statisticis computed, on the
values of variables TFPCH, ECH and TCH, given by table 5. Value of H in Hopkins test lies
between 0 and 0.5; close to zero means data is clusterable and close to 0.5 means non
clusterable.For the present data, H = 0.2768939, which indicates that the data is
clusterable.Further, to find the appropriate number of clusters in which the public sector banks
should be divided into, Hierarchical clustering is performed.Results suggest that taking five
number of clusters will be most appropriate. The dendrogram in figure 1 gives the detailed
division of objects into five clusters and genealogy of clusters.
For proper grouping of objects into homogeneous clusters, K-means clustering is done for k=5,
which gives five clusters C1, C2, C3, C4 and C5 of sizes 1, 12, 3, 4 and 5 objects
respectively.Cluster means of these clusters are given in Table 6.Geometrical representation of
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partitioning of DMUs in five clusters, byK-means, is given in figure 2. It is evident that DMUs
within each cluster form homogenous groups. Mutual distance in position of any two DMUs tells
about the extent of similarity in their efficiency behaviour. Information given by figure 2 is
summarized in Table 7.Table 7 shows the clusters in first column and the DMUs present in each
cluster are given in column two. Third column gives the number of years, out of total 16 years,
for each DMU, for which that DMU is found to be efficient in DEA analysis in step 1(Table4,
last column).Cluster C2 is the biggest cluster with 12 DMUs. But DMU 2 is the best performer
which has been DEA efficient in all 16 years. Thus, for all other DMUs in cluster C2, benchmark
is DMU 2. Cluster C3 has DMU 7 as its best performer. So, for DMU 12 and DMU 20,
benchmark is DMU 7. Similarly, in Cluster C4, DMU 16 is a benchmark for DMU 9, DMU 22
and DMU 24. Although benchmark DMU 16 itself is not showing a very good performance, but
even then, it is important for the DMUs in this cluster to consider it as a benchmark because
improvements are possible in a gradual manner by first achieving a feasible target.Likewise, in
Cluster C5, benchmark for DMU 3, DMU 6, DMU 10 and DMU 17 have their benchmark as
DMU 14. Cluster C1 has only DMU 15, for its benchmark,C1 can be considered as merged into
cluster C3, so, DMU 15 should follow DMU 7 as its benchmark.
V. Conclusion
From theexisting literature review, it is found that Data envelopment analysis (DEA) has been
used extensively for efficiency evaluation and as a benchmarking of DMUs,in a single time
period. Studies have also applied Malmquist TFP index on DEA results to analyse changes in the
efficiency levels, fromone time period to the next. But, none of these researches have given
benchmarks considering multi period data. Hence the present study proposes to add value to
existing literature by proposing a new Malmquist clustering benchmarking model based on DEA
(MCB DEA) to calculate benchmarks using cluster analysis with DEA based Malmquist TFP
index.
From the foregoing analysis and development of MCB-DEAmodel and further testing the same
on 25 public sector banks operating in India, for the time period from the year 1998 to the year
2013, efficiency of each DMU is evaluated by using CCR model of DEA for each year
separately. The result shows that during all these sixteen years under study, out of total 400
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observations, 204 results in efficiency and 196 observations indicate inefficiency.Then, the
change in efficiency behaviour of each DMU, over the entire time period, is analysed using
Malmquist TFP index, on the efficiency scores given by DEA. It is observed that over the entire
period of study, out of total 25 public sector banks, 20 have shown an increased TFP on average
annually and an improvement in technical efficiency is responsible for this increased
TFP.Further, on the basis of the overall Malmquist TFP index and its components, technical
efficiency change and efficiency change, public sector banks under study are divided into five
homogeneous clusters C1, C2, C3, C4 and C5. The best performing public sector bank in cluster
C2 is The State Bank of Bikaner and Jaipur, so it is the benchmark for all other banks falling in
cluster C2. Similarly, The Allahabad Bank is the benchmark in clusters C1 and C3 merged
together, The Indian Bank is the benchmark in cluster C4 and The Corporation Bank is the
benchmark in C5. Hence it is concluded that the benchmarks found on the basis of MCB-DEA
model are inherently similar to their respective less efficient banks, thus providing more realistic
targets to achieve, in order to improve efficiency gradually.
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Table 1: List of Banks under study Name of the Bank Group State Bank of India State Bank of Bikaner and
Jaipur State Bank of Hyderabad
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State Bank of Mysore SBI & Associates State Bank of Patiala State Bank of Travancore Allahabad Bank
Other Nationalized
Banks
Andhra Bank Bank of Baroda Bank of India Bank of Maharashtra Canara Bank Central Bank of India Corporation Bank Dena Bank Indian Bank Indian Overseas Bank Oriental Bank of
Commerce Punjab National Bank Punjab and Sind Bank Syndicate Bank UCO Bank Union Bank of India United Bank of India Vijaya Bank
Table 2: Description of Input and Output Variables
Variables Description
Input Variables Owned funds Sum of Capital and Reserves
Deposits Total deposits
Borrowings Total Borrowings
Wage Bills Salaries to all employees
Output Variables Spread Interest Earned Minus Interest Expended
Other Income Sum of income from Commission,
exchange & brokerage etc
Table 3: Descriptive Statistics of Input & Output Variables
N Minimum Maximum Mean Std. Deviation
Owned Funds 400 2177.0 988837.0 57197.83 98725.56
Deposits 400 47686.0 12027396.00 852906.29 1292143.55
Borrowings 400 2.0 1691827.0 51514.33 147160.26
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Wage Bills 400 1289.0 183809.0 11991.01 19325.68
Spread 400 0.0 443313.0 26146.32 44157.92
Other Income 400 522.0 160348.0 10831.73 18991.88
(All variables are measured in Million Indian Rupees.)
Source: Authors‟ own calculations
Table 4 : DEA Efficiency Scores (%)
YEAR
DMUs
1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Number
of
Efficient
Years
(out of
16)
State Bank
of India
95.4 100 83.8 80.6 87.3 81.9 83.6 88 100 94.6 100 100 100 100 100 100 8
State Bank
of Bikaner
and Jaipur
100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 16
State Bank
of
Hyderabad
100 100 100 100 100 100 100 85.8 89.9 100 100 95.9 100 100 100 100 13
State Bank
of Mysore
100 100 100 100 100 100 100 100 100 100 100 90.5 100 100 89.5 100 14
State Bank
of Patiala
100 100 100 100 100 100 100 100 94.7 89.5 100 91.8 99.5 100 96.9 93 10
State Bank
of
Travancore
100 100 100 100 95 100 100 100 100 100 98.5 100 100 89.9 93.5 94.6 11
Allahabad
Bank
100 100 100 95.9 100 100 100 100 100 100 98.2 100 100 94.2 95.8 88.7 11
Andhra
Bank
85.5 84 91.2 81.1 90.8 100 100 100 92 98.8 100 93.4 100 100 100 98.7 7
Bank of
Baroda
78.8 91.4 85.3 83.2 77.2 89.2 93.7 88.6 84.1 90 89.9 94.5 91.8 94.9 100 100 2
Bank of
India
79.3 83.2 76.3 80.5 89.2 94.1 88.7 72.7 86 95.4 98.3 100 88.9 82.8 91 95.7 1
Bank of
Maharashtra
85.9 88.9 95.2 100 100 93.2 81.4 75.8 96.5 100 100 99.3 95.6 89.4 97.8 95.2 4
Canara
Bank
88.1 94.9 82 84.2 100 100 100 100 100 91.3 100 87.2 100 96.9 100 85.7 8
Central
Bank of
India
90.4 84.6 74.7 75.2 94.3 100 100 100 100 100 100 78.7 94.8 84.9 77.4 74.1 6
Corporation
Bank
100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 16
Dena Bank 93.4 87.3 74.8 69.5 95.6 100 100 84.1 100 100 100 100 100 100 96.7 93.7 8
Indian Bank 59.9 60.6 62.1 67.4 82.7 79 96.2 88.6 89.6 100 100 100 100 100 100 100 7
Indian
Overseas
Bank
71.9 82.2 100 100 100 100 100 100 100 100 100 98.9 87.4 86.1 85 88.2 9
Oriental
Bank of
Commerce
95.5 100 100 100 100 100 100 100 100 100 99.6 100 100 100 100 100 14
Punjab
National
Bank
100 100 84.6 88.2 95.2 100 100 87.7 93.1 100 100 100 100 100 99 100 10
Punjab and
Sind Bank
75.4 81.4 75.3 88.2 89.5 100 100 100 100 100 94.1 91.1 90.5 78.7 66.6 79.8 5
Syndicate
Bank
96.5 100 100 100 100 100 98.3 91.8 99.9 96.7 91.5 93.2 97.2 100 100 96.2 7
UCO Bank 55.3 62.6 67.4 67.2 82.9 86.1 90.3 78.4 90 94 95.6 91.4 89.6 99.5 89.3 100 1
Union Bank
of India
88.8 100 58.6 84 100 94.9 87.8 87.2 100 100 100 100 100 89.7 100 96.4 8
United
Bank of
India
71.5 59.1 52.4 56.1 76.3 96.3 100 100 97.4 88.6 69 76.4 100 89.2 92 100 4
Vijaya
Bank
68.9 76.5 87.2 88.6 95.5 94.9 100 100 100 100 89.4 99.4 99.9 85.7 78.9 74.7 4
Total No. of Observations = 400; Efficient = 204 ; Inefficient = 196
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Source: Authors‟ own calculations
Table 5 : Average Annual Changes in TFP, Efficiency and Technical Efficiency of
DMUs DMU
Id
DMUs TFPCH ECH TCH
1 State Bank of India 1.023546 0.996863 1.026749
2 State Bank of Bikaner and Jaipur 1.02502 1 1.02502
3 State Bank of Hyderabad 1.012457 0.999953 1.01238
4 State Bank of Mysore 1.019448 0.999983 1.019478
5 State Bank of Patiala 1.025155 1.004858 1.02016
6 State Bank of Travancore 1.012633 1.003839 1.008757
7 Allahabad Bank 1.071444 1.008024 1.062966
8 Andhra Bank 1.012455 0.990429 1.022039
9 Bank of Baroda 1.008336 0.984391 1.024633
10 Bank of India 0.996524 0.987436 1.009203
11 Bank of Maharashtra 1.026028 0.993163 1.032957
12 Canara Bank 1.11351 1.001891 1.111482
13 Central Bank of India 1.026666 1.013373 1.013053
14 Corporation Bank 0.995872 1 0.996434
15 Dena Bank 1.176136 0.999873 1.1762
16 Indian Bank 0.993501 0.96631 1.028004
17 Indian Overseas Bank 0.992851 0.986533 1.006571
18 Oriental Bank of Commerce 1.033275 0.996934 1.036451
19 Punjab National Bank 1.028605 1.000001 1.028629
20 Punjab and Sind Bank 1.068782 0.9961 1.072766
21 Syndicate Bank 1.035742 1.000405 1.035484
22 UCO Bank 0.97865 0.961367 1.018124
23 Union Bank of India 1.021955 0.994439 1.027589
24 United Bank of India 1.010761 0.977845 1.033687
25 Vijaya Bank 1.027924 0.994605 1.033539
Source: Authors‟ own calculations
Table 6: Cluster means
Clusters TFPCH TCH ECH
C1 1.176136 0.9998730 1.176200
C2 1.025485 0.9987544 1.026762
C3 1.084579 1.0020050 1.082405
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C4 0.997812 0.9724782 1.026112
C5 1.002067 0.9955522 1.006669
Source: Authors‟ own calculations
Fig 1: Two-Dimensional Dendrogram showing partitioning and genealogy of clusters
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Fig 2: Two-Dimensional representation of K-means clustering results
Source: Authors‟ own calculations
Table 7: Identification of Benchmarks
Cluster DMU Id No. of years with DEA „efficient‟
Status (out of 16)
Benchmark
C1 15 8 DMU 7
C2
1 8
DMU 2
2 16
4 14
5 10
8 7
11 4
13 6
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18 14
19 10
21 7
23 8
25 4
C3
7 11
DMU 7 12 8
20 5
C4
9 2
DMU 16 16 7
22 1
24 4
C5
3 13
DMU 14
6 11
10 1
14 16
17 9
Source: Authors‟ own calculations